In the movie WALL-E, the Earth is made a garbage dump, Why are no other planets used as garbage dumps?
Earth is not being used as a Garbage Dump. Earth has BECOME a garbage heap due to decades of mass consumerism facilitated by the megacorporation Buy 'n' Large and is finally evacuated by the same company. That is why Wall E and other units are employed in the Operation Cleanup.
First, the canon answer. There's nothing in the movie to suggest they didn't try this. The very dense ring of space junk, the Buy N Large billboard on the Moon, and the size and technology of the Axiom suggest they had very, very active commercial space flight when Humanity abandoned Earth.
Next, the social answer. Whether or not it was technically possible (which I'll get into below), Wall*E is an allegory for the modern day disposable society. It's about creating sympathy for a problem, not about proposing solutions. Within the dystopia presented by the movie, as well as the world now, the real answer is... complicated. If you want a detailed examination of this problem, read "Collapse: How Societies Choose To Fail Or Succeed" by Jared Diamond which answers the similar question "what was going through the mind of the person who cut down the last tree on Easter Island".
Now, the technical answer. They could do it, but it wouldn't be practical. It would require a significant fraction of the world's energy to do it. There's a lot of trash and it would take a lot of energy to get stuff off Earth. Here's the math (I know Wall*E isn't hard sci-fi, but people think we can just shoot garbage into the Sun so I'm going to run the numbers anyway).
You need 11.2 km/s to escape Earth's gravity. To achieve that for 1 kg of mass, ignoring all considerations but gravity, takes about 62 MJ or about half a gallon of gasoline. Drag, fuel, inefficiencies and space craft mass would make that take many times more, but Wall*E demonstrates some pretty amazing spacecraft so let's just go with an absolute lower bound. Maybe they have a space elevator or something.
That's just to escape the Earth. Reaching another planet, even one closer to the Sun, requires even more energy. Mars requires about 25 km/s which adds six times the energy (kinetic energy is velocity squared). Shooting it into the Sun would require even more energy. The garbage doesn't have to go to another planet, it just has to get off the Earth and stay off it. Let's place it at the stable Earth-Sun L4/L5 Lagrange points which don't require much more than Earth escape velocity to reach. It can just float around in Solar orbit like the Great Pacific Garbage Patch.
How much energy would we need to shoot current mass of trash off the Earth? The best estimate for how much trash we produce I could find is 1 billion tons per year. Shooting that off Earth requires at an absolute, unrealistic minimum 5.7 x 10^19 J per year or about 10% of the world's energy consumption. A realistic program including transporting the trash to launch sites, atmospheric drag, engine inefficiencies, etc... would likely require all the world's energy.
That's just the present day numbers. The Earth we see in Wall*E was producing far, far more trash than that in 2100. Even with advances in space flight and energy production, it's just not feasible. However, the absurd density of space junk EVE's probe and the Axiom had to break through suggests a solid shell of trash now surrounds the planet representing a very impressive investment of energy and material. Someone else can run the numbers on that.